Excellent News, good for you turning a challenge into a really refreshing opportunity.
as for the name, I think here are two camps here...either something really odd that sticks in the mind like...3...or Google...or
Ebay...or....Facebook, you can see the effect
Personally as a new, small business with no intentions on world domination (which is where these anonymous names really help as they don't restrict the firm to a location) I would try to make it as short as possible but allow it to describe in a nutshell what you do...so its a no brainer.
Also, be warned about allowing too much of your personality into the name (unless you are a really well know local personality) as this will intimidate the less outgoing folk from using you...keep it professional and a little dull.
if you make it funny, I'd try to make the joke appeal across all the ranges, not just those in the know.
Some ideas...
Homeserv... Mobile bike servicing (professional, dull and descriptive and inoffensive)
Mobile-Monkey...bringing the grease monkey to your doorstep (funny, informal and a great opportunity for a logo...although you may get bookings for kids party's)
Two wheels-to door. Mobile Motorcycle servicing...bit naff and clumsy
Bike Doctor -Mobile motorcycle servicing (I'm pretty sure the whole "Doctor" thing has been done to death but maybe not in the mechanics world)
Service 2 Go-Mobile motorcycle servicing
Fluid- Flexible home servicing
Mobile engineering-Mobile motorcycle servicing
Fine-Tune...Mobile motorcycle servicing
or put your name on the side of the van/ape/truck/bike and call it Steve Smith (Insert real name) mobile motorcycle servicing.
Perhaps